A prior probability is the probability assigned to an event before additional evidence is considered. In Bayes’ rule, it is written as P(A); evidence enters through a likelihood, and the updated probability is the posterior P(A|B). section 12.2
For an unknown parameter θ, a prior can be a full probability distribution rather than one number. Bayesian updating combines that distribution with data to form a posterior distribution, treating θ as uncertain rather than fixed. section 12.2
Prior returns in Black–Litterman
Expected asset returns begin with a multivariate normal prior having mean μ and covariance Σ + C. Here, C represents uncertainty in expected returns; a common convention sets C = τΣ, preserving the return-correlation structure while scaling uncertainty. section 12.3.3 section 12.3.1
Investor views supply uncertain evidence: AR is modeled as normally distributed with mean b and covariance Ω. Updating the prior with these views yields a normal posterior mean and covariance, which can then feed standard asset-allocation optimization. section 12.3.3 section 12.3.2
Listing 12.1 Posterior mean and covariance
import numpy as np
import pandas as pd
def compute_posterior_mean(mu, C, A, b, omega):
c_inv = np.linalg.inv(C)
a_t_oinv = A.T @ np.linalg.inv(omega) #1
er = np.linalg.inv(c_inv + a_t_oinv @ A) @ (c_inv @ mu + a_t_oinv @ b)
return pd.Series(er, index=mu.index)
def compute_posterior_cov(sigma, C, A, omega):
return sigma + np.linalg.inv((np.linalg.inv(C) + A.T @ omega @ A))
Use the posterior-mean and posterior-covariance functions to implement the normal-distribution update for general C, A, b, and Ω inputs. section 12.3.3
